No‐(k+1)‐in‐line problem for large constant k

Source abstract

Abstract How many points can be placed in an grid so that every (affine) line contains at most points? We prove that for the maximum number of points is exactly . Our proof builds on the recent work of Kovács, Nagy and Szabó (who proved an analogous result when is at least about ), incorporating ideas of Jain and Pham. Using the same approach, we also obtain new bounds for higher‐dimensional extensions of this problem.